Chapter 28
PRICING OF FUTURES
AND OPTIONS
CONTRACTS
Arbitrage Strategies
Cash and carry trade
borrowing cash to purchase a security and
carrying that security to the futures
settlement date
Reverse cash and carry trade
selling a security short and investing the
proceeds received from the short sale
Pricing of Futures
Contracts
The theoretical or equilibrium futures
prices is based on arbitrage arguments.
The following information is needed:
The price of the asset in the cash market
The cash yield earned on the asset until the
settlement date
The rate for borrowing and lending until the
settlement date
A Theory of Futures
Pricing
The equilibrium futures price is the price
that ensures the the profit from the
arbitrage strategy is zero.
Profit = 0 = F + yP - (P + rP)
The theoretical futures price is:
F = P + P (r - y)
A Theory of Futures
Pricing
The theoretical futures price depends on:
The price of the underlying asset in the cash
market.
The cost of financing a position in the
underlying asset.
The cash yield on the underlying asset.
Theoretical Futures Price
The effect of carry on the difference between
the futures price and the cash price can be
shows as follows:
Carry
Futures price
Positive carry
(y > r)
Will sell at a discount to
cash price (F < P)
Negative carry
(y < r)
Will sell at a premium to
cash price (F > P)
Zero (r = y)
Will be equal to the cash
price (F = P)
Principle of Convergence
At the delivery date, the futures price
must equal the cash price.
As the delivery date approaches, the
futures price converges to the cash price.
The financing cost approaches zero
The yield approaches zero
The cost of carry approaches zero
Assumptions Underlying
the Arbitrage Arguments
Interim cash flows
Differences between lending and borrowing
rates
Transaction costs
Short selling
Known deliverable asset and settlement date
Deliverable is a basket of securities
Different tax treatment of cash and futures
transactions
Pricing of Options
The price of an option consists of two
components: the intrinsic value and the time
premium.
Intrinsic value
the economic value of the option if exercised
immediately, which is either greater than zero or zero
Time premium
amount by which the option price exceeds the
intrinsic value
Put-Call Parity
Relationship between the price of a call,
and the price of a put
On the same underlying asset
With the same strike price
With the same expiration date
Put-Call Parity
Put-call parity for European options with
cash distributions on underlying asset:
X  Dt
PC 
S
t
(1  rf )
where: P = Put option price
C = Call option price
X = Strike price
Dt = Cash distribution
S = Price of underlying asset
rf = Riskfree rate
Factors That Influence the
Options Price
Current price of the underlying asset
Strike price
Time to expiration of the option
Expected price volatility of the underlying asset
over the life of the option
Short-term, riskfree interest rate over the life of
the option
Anticipated cash payments on the underlying
asset over the life of the option
Option Pricing Models
The theoretical options price is
determined on the basis of arbitrage
arguments.
Option Pricing Models
Black and Scholes Option Pricing Model
Binomial Option Pricing Model
Binomial Option Pricing
Model
Hedged Portfolio
Long position in a certain amount of the asset
Short call position in the underlying asset
Cost of Hedged Portfolio
HS - C
Payoff of Riskless Hedged Portfolio
uHS - Cu =dHS - Cd
Hedge Ratio
H = (Cu - Cd)/(u - d)S
Price of a Call Option
Hedged Portfolio
HS - C
One-Period Wealth
(1 + r)(HS -C)
Payoff of Hedged Portfolio
uHS - Cu
Call Option Price
 1  r  d  Cu   u  1  r  Cd 
C 




 u  d  1  r   u  d  1  r 
Assumptions of Binomial
Model
Price of the security can take on any
positive value with some probability
Short-term interest rate is constant over
the life of the option
Volatility of the price of the security is
constant over the life of the option
Fixed-Income Option
Pricing Models
Assumptions of binomial model are
unreasonable for fixed-income securities
Alternative option pricing models:
yield curve option pricing models
arbitrage-free option pricing models